Hermite-Birkhoff interpolation problems in Haar subspaces

Hermite-Birkhoff

Interpolation

DEDICATED ON

Problems

TO PROFESSOR

‘THE

OCCASIOiX

I.

in Haar

Subspaces

I. J. S(‘HOT;NtWRG

OF HIS

70TH

RIKTHDA’I

INTRODLKYIO~

Let E = (E,,). i --: I...., /),,j - 0, I,..., y, be a rectangular matrix with /) 0) such that each element of it is rows (p I) and (/ -I I columns (q either 0 or 1. Such a matrix is called an it~ci&~rc~~ matri.y. Let II be the total number of I’s in E.

In the sequel we assume 1 _ 11 II I Let H be an n-dimensional subspace of (“‘(a. h), the space of (i-times continuously differentiable real functions defined on an open interval (a, 6). Select arbitrarily a set of p nodes, x1 6. x2 ‘.. .Y,, . from (a. b). Further. I. select an arbitrary number corresponding to each (i,,j) for which E,, a’,’ ‘.

We consider the following HermiteBirkhoJjf interpolation a function I7 E H which satisfies the jr interpolator-y conditions:

whenever

We shall say that E is poised with rrspect ;I unique solution to the above problem. .yL . .. .. s,, and the numbers ajJ’. ’ \Vork on Army Research attarded

to The

this paper was Office, Durham, University

supported and

of Texas

I:

E,,

by

I42

I . 2.. . 1’.

j

Find

0. 1 . .. q.

if there always exists of the choice of the nodes

to f! u/i (a, h)

regardless

in part National

at Austin.

i

prohletll.

b> Grant Science

L>A-ARO(U)-31-124-(;1050. Foundation tirant

CiP-23655

HERMIT&HIRtiHOf~F

I4TERPOLATIO\.

143

We now mention a number of relevant papers related to our main result. which is given in the next section. G. P6lya [5]. I. J. Schoenberg [7], and K. Atkinson and A. Sharma [I] consider the case where H -:- ?T,, , , the space of all polynomials of degree at most II I. The results obtained by the latter authors contain ours for this special case. I. J. Schoenberg assumes a Her-mite-type condition on E. that is. for i -~ I,.... 11. E,,,. L I implies I. On the other hand, J. W. Matthews [4] proves our main theorem E,,, for the case in which E consists of only two columns (r/ ~:mI) and satisfies the above Herrnite-type condition. W. Haussmann [2], generalizing Matthews’ result but still assuming the Hermite-type condition on I:. provides a simpler proof for Matthews’ result and proves further that. if dim H =~ dim H(l)

== n(H”)

{ril~dx

: II E HJ),

and both H and H”’ are Haar subspaces, then E is poised with respect to H on (a. b). Our main theorem does not include this last fact. even though both proofs are based on the same known theorem quoted in Sec. 3.

2.

Let H be an /~-dimensional dim H”‘)

MAIN

THEOREM

subspace

L= dim{/lf”)(.u)

of C”(a. h) and satisfy the condition

: /I E Hj ~~~II

(1

(q

n -- I)

and let NC”) satisfy the Haar condition on (a, h), i.e.. every nonzero function in HI”’ vanishes at dim H”” - I or less distinct points on (a, b). Then. in order that E be poised with respect to H on (a, 6) it is sufficient that E satisfies the Pdl)sa condition and is consrr~atice. We say that E satisfies the P6lya condition if M,,

I. M,

3 .-...., M,,

l/-

I.

where M, 7 n1,, f

111,

.‘I ~+ 111,

E,, z= the number

(j == 0. I ,...~ y). of I’s in the,jth

column

of E.

In order to define the conservativeness of E, let us first define a sey~rrrce in E as in [I]. A sequence in 15 is a maximal sequence of I’s in any row of E. For instance, the matrix

144

IKEBE

has a sequence {I, 1) starting at (I. I )-position. It has seven sequences in all. For convenience, if a sequence starts at (i,j)-position. we call it the (i. j)seque~e. A sequence is et~/r or odd depending on whether the number ot I’s in this sequence is even or odd. We say that E is conservative if an arbitrary sequence (let it be (i,, , j(,)-sequence) is either even or else the upper left portion of E defined by (E,, : i I. i,, and j -.. j,,; and the lower left portion of E defined by {E,, : i

i,, and j < ,j,,i

do not simultaneously contain I’s, One or both of these sets may be empty. For instance, the last matrix E is not conservative since the (3. 2)-sequence { I _ I. I j is odd and the upper left portion of E and the lower left portion of E defined in the above simultaneously contain 1’~ (there are two I’s in the former and one I in the latter). The matrix

E;~

0 I 1 I 0 0 lOllII 0 I I 0 I I 0 0 0 0 I 0

is conservative. Note that if L.’ satisfies the Hermite-type condition mentioned in the last section then E is automatically conservative. Before proceeding to the proof. we note a few simple consequences of the hypothesis of the main theorem. They arc (a) dim I/‘)) 47 i(i 0, I ,.._, y), where HI” (b) H”” H. H(” I . H(‘l’ are all Haar subspaces: (c) H contains polynomials I, .Y..__..V ‘.

iIl”‘(.Y) 1 h C Hi:

To prove (a). it is enough to prove that dim I-I”’

I

dim H1""

dim H’” (i

The proof of this last fact is easy and is omitted. theorem and (a). Part (c) follows easily from (a).

3.

PKOOI

OF ~‘HF

MAIN

o..... (1

I).

To prove (b). use Rolle‘s

THEOREM

In order to prove the main theorem, it is necessary and sulhcient to prove that the homogeneous system obtained from (I) has only the trivial solution h =:=0. Thus let 17t If satisfy (I) with all aj”‘s 0. In order to prove /I --= 0, it is

I-IERMITL-BIRICHOFF

145

INTtRPOL.ZTlOh

enough to prove that /I’,,’ ~- 0 and h”’ vanishes on (a, h) at least once for 0, it is enough to prove that /I”~’ i 0. I ,..., y ~~ I. In order to prove /I(“) vanishes at least 17 I/( ---dim H”‘)) times on (a, h). counting each ~‘ozrhk :rro twice, on the strength of the following theorem. THEOREM.

([3, Theoretic

m-dimetzsionulHuur sd~spuce [u. h] at least 177 tit1w.s. eo7777tiq

4.2, p. 231 and [6, Lemnm 3-2, 11. 571) /f Ail is un of‘ C[u, h] ut7d if’ a ,/iwc.tio/?,f’ in M ~~unishes OII each

double

zero

tw’ice,

th

0 017

f

[a.

h].

An interior point c of [a. h] is called a situp/e ;UO (resp., a double zero) of ~IF C[u. h] if,f(c) =-: 0 and if,f’changes (resp.. does not change) sign at c’ in the sense that 0 (resp.,f(t,)f(t,)

.f‘(t,).f(tJ

0)

whenever t, L ((a - 8. C) and t, t (c, c - 6) with a sufficiently small but fixed b) andf(c) -= 0, then c is a simple (3 0. If c is an endpoint (c ~= u or (7 zero by definition. Let us write E,, for E to emphasize the fact that the incidence matrix E is associated with the function /I. In the sequel we shall construct. by induction. a sequence of incidence matrices E:$, = E,( . E:;:/, ..... E:,:‘b,, where EL::, denotes an incidence matrix associated with /I~/‘, i 0, I...., q, in such a way that the following properties are satisfied: (i)

Eilj, has one less number

(ii)

E:,I!, has at least

colw77~ zero

(and qf’

(iii)

oi7ly

in

its

17

of columns

i I’S,

right-most

where

column)

we twice

than

&IT ‘i, ,

count if‘

I

each that

i

I.....

it7

its

1 it7ciicatr.r

q:

right-nmst u double

I7 (“I.

Ei:!, satisfies the Polya condition

and is conservative,

i

= O..... ~1.

If the indicated sequence of matrices is in fact constructed, then, by (i), the right-most column of EL:!, (i 1 O,.... q) always corresponds to the function /rI”) and the matrtx E,,,,,, (“I consists of a single column. Then. by (ii), k(‘f) vanishes at least II ~ ~1 times, counting each double zero twice. Property (iii) implies that /I(~), i =~ O,..., 4 ~~ I, vanishes at least once on (a, b). Thus, the proof will be complete if we construct the indicated sequence {E~~$j satisfying properties (i)-(iii) in the above. The incidence matrix EL?, = E,, satisfies the indicated properties by definition. where property (i) is satisfied vacuously. Let

A’, - (xi : E,,,

_ I ),

,; = 0. I . .. .. c/:

that is, /I”‘(.x,)

-= 0

for

.Y, c x,

By the POlya condition, the left-most column of E,, contains at least one 1. Hence A’,, is not empty. If E,, contains exactly one I. let C-:,::, be defined to be that matrix which is obtained from E,, by deleting its left-most column. l-hen. E:,::, clearly satisfies properties (i)-(iii). Thus. suppose that E,, contain\ ? or more I’s in its left-most column. Let X,) contain /, points. I, 1, .. 1, Hence lz(tt) “. 0. T-&e the interval 11, . l:+] and let I? take hf,, 1 the maximum on it at 5, .

We may assume t, < tJr c I,. Obviously. hiiJ(l;,) 0. I lere two separate cases may arise: (a) 5, 4 X, and (b) [, C: X, If (a) is the case, then the fact that /I”)(<,) 0 is new information ;Ihout It”’ which is nut indicated by E,, For instance. if

If (b) is the case. el .Y,,] I_ X, for some i,, By the hypotheses that “‘<_ t,, is the complete enumeration of the nodes appearing in X,, and by the fact that I, I 5, i t, . the node .Y,,, does not appear in A’,, This means the existence of a (i,, , I)-sequence in E,, For instance. if

t,-:t,.

E,,

then X,, - (s, . .v:,) and X, =- (.Y, . .u.,]. It may happen that <, Now, since E,, is conservative. the last (i,, . ! )-sequence must this sequence consist of 117 I’s (UT, ever1 1. Then

.x’:!(i,, be even.

2). L.et

from the fact that h takes a local extremum at [, . it follows that <, is a ,sinrp/c zero of/?“). It then follows that 5, is a double zero of /I@). hC4),.... /I(“” and is a simple zero of hC3),..., h+ lj (we omit the easy proof of this fact). Since E, is a double zero of /I(“‘), it is a simple zero of hctrL I) provided that 17”“) t Cl. In terminates before the other words, if the (i, , I)-sequence under consideration I at the (i,, , tn I)right-most column of E,, , we can write an additional position of E,, . Otherwise we have n7 = L/ and 5, is. therefore. a double zero of htq’.

HERMITE-BIRKHOFF

We can carry [I, , r,] ,..., [tkpI , MJ = mqtp,t31 incidence matrix

INTERPOLATION

147

out the same analysis for the rest of the subintervals fk]. Let t2 ,..., 5/;-l play similar roles as t1 , i.e., t, i 5, < t, , / k / , etc, Using these 5, ,..., tLpl ? we can construct an E$, for h(l) m an obvious way. For instance, if

then A’,, = {x, , x3, x4} (tl = X, , t, =-: x3 , t, = x4) and XI =- {x2}. The matrix Ejl is conservative and satisties the P6lya condition. Thus, if
where x1 < x2 < El < x3 (here, of course, the case ?cr < e1 < X, < x3 may occur. in which case we can construct E$ similarly). If [r = x2 , then 1 I 1* E(1) ~ 1 0 I 1 1 A(‘) 1 0 0 -0 0 1

... ... .” ..’

x., 53 5, ’ x,

where ,$I = x2 and the 1 with an asterisk (*) is new information For yet another case, take E, ::

(/zC3)(x2) =- 0).

100.‘.x, i 1 0 I 1 0

I ... x2 0 ..’ X3

If fl -= x, (x, < 5, < x,), then we take &:I), to be E$, ==: [l I*] .‘. xp ) where the 1 with an asterisk (*) indicates that x2 is a double zero of h(‘). From the construction of E:L:!j , we can verify without difficulty that conditions (i)-(iii) are always satisfied for i : 1. In fact, verification of (i) is trivial. In order to verify (ii) we note that for each pair of consecutive l’s in the left-most column of EI, , a new entry 1 is introduced in E$ in addition to those l’s m EL:!, which are already present in E,b . In order to verify the

conservativeness of E,LCl, (‘I we need only to check those sequences in Ej,::, which do not originate in the left-most column of E:L::r . Each of these sequences is, therefore, already present in IL’,,. except perhaps its last term. From thih consideration it is now clear that E”’,iCI, IS conservative. There remains only to show that E:,:il satisfies the P6lya condition. Suppose the contrary. Then there would exist a j,, such that

but

/ll,,(E,::I,,

“’

lll~,,(E~~~~i

i,,

I.

where III,(E;,~I,) denotes the number 01‘ I’> in the j-th column of 15:~:: j ~-= 0, I,.... (1 I. This means that the number of the I’s in L$,:!, contained to the left and including the ,j,,-th column of E;
which is a contradiction. Starting with E,,(I, (I’ , we can similarly f y conditions (i)-(iii). This completcb

construct the proof.

6;;,;:, ._... Eb;‘a) Mhich all satis-

Under the hypothesis of the main theorem we can prove that the Ptilya condition is necessary in order that E is poised with respect to H on (CI, h). The proof runs in a similar line as in [7. p. 5401. In fact, if the Phlya condition is /JIM + ... I /in,,, j,) i- I with not satisfied then there is a,j,, such that 177,~ in H (see Sec. 2). so is silo It is possible to .I0 *\’ (I. Since r,,~, is contained find a nontrivial polynomial h(s) (‘,.Y : .” + c,~,x’[~ which satisties (‘0 the homogeneous system (I), because, for ,j ‘,jO /I(~) m= 0 is automatically satisfied and for,j I of unknowns (c,, .. . .. c,,,) exceeds the ,j,, the number,j,, number n?,, -1~ ... myN,~ of equations.

HERMITEEBIRKHOFF

INTERPOLATION

149

REFERENCES AND A. SHARMA, A partial characterization of poised Hermite-Birkhoff problems, SIAM J. Numr. Am/. 6 (1969), 230-235. On interpolation with derivatives, to appear in SIAM J. iV~mer. And. 2. W. HAUSSMANN, 3. S. J. KARLIX AND W. J. STCJDDEN,‘Tchebycheff systems,” Interscience, New York, I.

K. ATKINSON interpolation

1966.

J. W. MATTHEWS, Interpolation with derivatives, SIAM Ku. 12 (1970), 1:!7--218. 5. C. Po~ra, Bemerkung zur Interpolation und zur Naherungstheorie der Balkenbiegung, 4.

Z.

Atrgew.

Math.

Mrch.

11 (1931),

445449.

6. J. R. RICE, “The approximation of functions,” Addison-Wesley, Reading, Mass., 1964. 7. 1. J. SCHOENBERG,On Hermite-Birkholf interpolation, J. Murh. Appl. 16 (1906), 538-543.